Presently, a vibrating U-shaped-tube densitometer, where the natural mechanical frequency of vibration of the fluid-filled tube changes with varying fluid density is widely used to measure fluid density. A fluid is placed inside a vibrating U-tube and its resonance frequency is monitored. This frequency is related to the fluid density. Electromechanical elements and a feed-back loop amplifier maintain the vibrations and provide a frequency output determined by the fluid density. Such measurements require that fluid from a flowing pipe is diverted into the U-tube.
Using a quasi-steady and homogeneous fluid model, the mass of the fluid effectively adds to the mass of the U-tube/fluid system since the fluid typically has little effect on the stiffness of the system. Introducing fluid into the tube then changes the natural frequency of the oscillation of the system. The mass of the fluid in the tube is proportional to fluid density, ρfluid, and the natural frequency, fnat, of the system decreases with increasing fluid density in accordance with:
            f      nat        =                  1                  2          ⁢          π                    ⁢                                    K            struct                                              m              struct                        +                          β              ⁢                                                          ⁢                              ρ                fluid                                                          ,where, β is a calibration constant related to the geometry and vibratory characteristic of the vibrating tube, mstruct is the mass of the tube structure, and Kstruct is a stiffness constant related to the elastic properties of the tube material. The natural frequency of the fluid loaded tubes can be further simplified as:
            f      nat        =                  f        s                              1          +                                    m              fluid                                      m              struct                                            ,where mfluid is the mass of the fluid inside the tube.
As stated hereinabove, the resonance frequency of a pipe or a cylinder (or any shape container) changes with fluid loading. The resonance frequencies for an empty pipe and a fluid-filled pipe can be expressed as:
            f      empty        =                  1                  2          ⁢          π                    ⁢                                    K            pipe                                m            pipe                                ,while
            f      full        =                  1                  2          ⁢          π                    ⁢                                    K            pipe                                              m              pipe                        +                          m              fluid                                            ,which can be rearranged such that the effect of fluid density inside a pipe can be expressed in terms of the natural frequency of the pipe as follows:
                    f        full            =                                    f            empty                                              1              +                                                m                  fluid                                                  m                  pipe                                                                    =                              f            empty                                              1              +                                                K                  pipe                                ⁢                                  ρ                  fluid                                                                          ,    where                      K        pipe            =                        V          pipe                          m          pipe                      ,    and  mfluid is the fluid mass, mpipe is the mass of the pipe or cylinder, ρfluid is the density of the fluid and Vpipe is the internal volume of the pipe, ffull is the frequency of a pipe filled with fluid, and Kpipe is a constant related to the elastic properties of the pipe material. The subscripts refer to the fluid and the pipe. Effectively, the above procedure determines the weight of a pipe with and without a fluid inside by monitoring its natural vibration frequency, and the density may be obtained from the mass.
The commonly practiced procedure requires installing a branch in the pipe bearing the fluid to be investigated, and attaching a vibrating U-tube or a Coriolis type meter external to the pipe. This necessitates the undesirable drilling of holes in the pipe, attaching flanges and other modifications of the pipe, all invasive procedures.
In U.S. Pat. No. 6,053,041 for “Noninvasive Method For Determining The Liquid Level And Density Inside Of A Container” which issued to Dipen N. Sinha on Apr. 25, 2000 describes a noninvasive method for determining fluid density by generating a flexural acoustic wave the wall of a container using ultrasonic tone bursts, and measuring the phase difference of the detected flexural wave from that of the originally generated wave a small distance from the generated wave, the magnitude of the phase difference being related to fluid density immediately opposite the measurement position on the surface of the vessel.